Lower bound the T-count via unitary stabilizer nullity

We introduce magic measures for multi-qubit quantum gates and establish lower bounds on the non-Clifford resources for fault-tolerant quantum computation. First, we introduce the stabilizer nullity of an arbitrary multi-qubit unitary, which is based on the subgroup of the quotient Pauli group associated with the unitary. This unitary stabilizer nullity extends the state stabilizer nullity by Beverland et al. to a dynamic version. We in particular show this magic measure has desirable properties such as sub-additivity under composition and additivity under tensor product. Second, we prove that a given unitary's stabilizer nullity is a lower bound for the T-count, utilizing the above properties in gate synthesis. Third, we compare the state and the unitary stabilizer nullity, proving that the lower bounds for the T-count obtained by the unitary stabilizer nullity are never less than the state stabilizer nullity. Moreover, we show an explicit $n$-qubit unitary family of unitary stabilizer nullity $2n$, which implies that its T-count is at least $2n$. This gives an example where the bounds derived by the unitary stabilizer nullity strictly outperform the state stabilizer nullity by a factor of $2$. We further connect the unitary stabilizer nullity and the state stabilizer nullity with auxiliary systems, showing that adding auxiliary systems and choosing proper stabilizer states can strictly improving the lower bound obtained by the state stabilizer nullity.